First - looking for already published research projects that contain and/or utilize datasets similar to what your project requires might be a good start.
There are some researchers and research groups that wouldn't have any issues sharing their dataset with you if you provide them with a good enough reason as to why you need it and what you're going to use it for.
That being said, because of privacy and data protection laws; it greatly depends on how they acquired the data in the first place and whether or not there is personally identifiable information within the dataset. If you don't do this right, you can get into quite a lot of trouble.
There are also a number of sites that supply anything from large national and international gov datasets to the more modest limited access sites.
Most U.S. government agencies provide open access to a large portion of their data.
The FBI's web services
The US Census Bureau
... the list goes on.
Kaggle is often talked about
BuzzFeed, surprisingly, provides a vast array of open information
... to name a few.
You may also wish to contact universities and college departments. The often work with massive datasets and are quite liberal when it comes to sharing.
Good luck :)